Rational/irrational numbers and simple number proofs | Year 11

01 / LEARN

The idea, step by step

Rational numbers are ratios of integers

A real number is rational when it can be written as $a / b$ with integers $a, b$ and $b \neq 0$ . The denominator restriction is essential. Integers, terminating decimals and recurring decimals are rational.

To prove a sum or product rational, combine the fractions and check that the new numerator and non-zero denominator are integers. Irrational numbers are real numbers that cannot be expressed as such a ratio.

Do not apply closure rules to irrational numbers

The sum of two irrational numbers may be rational or irrational. For example, opposite irrational numbers sum to zero, whereas a non-zero rational multiple of an irrational number stays irrational.

For a square-root irrationality proof, first choose a fraction in lowest terms. If the argument forces both numerator and denominator to share a factor, it contradicts this choice. This proof is about all possible fractions, not the decimal display of a calculator.

A common trap

A non-terminating calculator decimal proves irrationality.

Instead: A finite display cannot establish whether the exact decimal eventually repeats; use an exact argument.

02 / WORKED EXAMPLES

See the thinking

Example 1

A rational shift of an irrational number

Prove that $3 + \sqrt{2}$ is irrational.

Suppose $3 + \sqrt{2} = r$ is rational.
Then $\sqrt{2} = r - 3$ would be rational, because the difference of rational numbers is rational.
This contradicts the irrationality of $\sqrt{2}$ .

3 + \sqrt{2}

is irrational.

Example 2

The square root of two

Prove that $\sqrt{2}$ is irrational.

Suppose $\sqrt{2} = a / b$ in lowest terms, where $a, b$ are integers and $b \neq 0$ .
Then $a^{2} = 2 b^{2}$ , so $a^{2}$ is even. An odd integer has an odd square, so $a$ must be even.
Write $a = 2 k$ . Substitution gives $b^{2} = 2 k^{2}$ , so $b$ is also even.
Both share a factor of 2, contradicting lowest terms.

\sqrt{2}

is irrational.

Check your understanding

Is $\sqrt{2} \times \sqrt{2}$ irrational?

Show answer

No; the product is 2.

Two irrational factors can have a rational product.

03 / WORKSHEET

Your turn

6 questions · 17 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Prove that the sum of two rational numbers is rational.
2 marks
Prove that a non-zero rational multiple of an irrational number is irrational.
3 marks
Show that $0. \bar{27}$ is rational.
3 marks
Give irrational numbers whose sum is rational, and another pair whose sum is irrational.
2 marks
Prove that $\sqrt{5}$ is irrational.
4 marks
Is the reciprocal of a non-zero irrational number irrational? Prove your answer.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Prove that the sum of two rational numbers is rational.
2 marks
$a d + b c$ and $b d$ are integers and $b d \neq 0$ .
$a / b + c / d = (a d + b c) / (b d)$ .
Prove that a non-zero rational multiple of an irrational number is irrational.
3 marks
Let $r \neq 0$ be rational and $x$ irrational. If $r x$ were rational, then $x = (r x) / r$ would be rational, a contradiction.
The product is irrational.
Show that $0. \bar{27}$ is rational.
3 marks
Let $x = 0.2727 \dots$ . Then $100 x - x = 27$ , so $x = 27 / 99 = 3 / 11$ .
$3 / 11$ .
Give irrational numbers whose sum is rational, and another pair whose sum is irrational.
2 marks
0 is rational. A non-zero rational multiple of $\sqrt{2}$ is irrational.
$\sqrt{2} + (- \sqrt{2}) = 0$ ; $\sqrt{2} + \sqrt{2} = 2 \sqrt{2}$ .
Prove that $\sqrt{5}$ is irrational.
4 marks
Assume $\sqrt{5} = a / b$ in lowest terms. Then $a^{2} = 5 b^{2}$ .
A square divisible by the prime 5 has its root divisible by 5. Set $a = 5 k$ .
Then $b^{2} = 5 k^{2}$ , so 5 also divides $b$ , contradicting lowest terms.
It is irrational.
Is the reciprocal of a non-zero irrational number irrational? Prove your answer.
3 marks
If $1 / x$ were rational and non-zero, its reciprocal $x$ would be rational, a contradiction.
Yes.

Checking saved progress…

Continue learning

Interval notation

Next lesson

Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: geometric reasoning and real-number/proof material, regrouped into a school chapter; all exercises are original.

Official curriculum source ↗

Back to Proof and circle geometry